Home›Calculus 3›Equation of a plane›Problem 9.34

Equation of a plane

Problem 9.34 · hard

Find an equation of the plane through the points \( \displaystyle (1, -2, 1),\ (-1, 1, 2),\ (-1, 3, 2) \).

\[ \left[\begin{matrix}-2\\3\\1\end{matrix}\right] \]
PQ.
\[ \left[\begin{matrix}-2\\5\\1\end{matrix}\right] \]
PR.
\[ \left[\begin{matrix}-1\\0\\-2\end{matrix}\right] \]
A normal vector n = PQ × PR (scaled).
\[ -3 \]
n · P gives the constant.
The plane is -x - 2*z = -3.

Answer \( - x - 2 z = -3 \)

Lines: 5 not checked. The answer was also checked a second way, without looking at the solution. The reviewers disagree about how one step is explained; every verdict is in the receipt. How we verify

The full receipt

Line	Status	Checked by	Detail
1	Not checked	—	both sides are the same expression: the line states a value, and the working that produced it is not on the line
2	Not checked	—	both sides are the same expression: the line states a value, and the working that produced it is not on the line
3	Not checked	—	both sides are the same expression: the line states a value, and the working that produced it is not on the line
4	Not checked	—	both sides are the same expression: the line states a value, and the working that produced it is not on the line
5	Not checked	—	a sentence; read, not computed
answer, a second way	✓ Checked independently	sympy 1.14.0 + mpmath 1.3.0	all three points satisfy the equation

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: inconclusive — reviewer returned a non-object

Every verdict on record (6)

qwen3.6:27b-mlx: inconclusive 2026-09-28 — reviewer returned a non-object
gpt-oss:20b: pass 2026-09-28
qwen3.6:27b-mlx: fail (error) 2026-09-27 — The vector calculations are incorrect. The vector PQ should be (-2, 3, 1) but PR is calculated as (-2, 5, 1) instead of the correct (-2, 5, 1) wait, let's recheck. P=(1,-2,1), Q=(-1,1,2), R=(-1,3,2). PQ = Q-P = (-2, 3, 1). PR = R-P = (-2, 5, 1). The cross product PQ x PR = (3*1 - 1*5, 1*(-2) - (-2)*1, (-2)*5 - 3*(-2)) = (3-5, -2+2, -10+6) = (-2, 0, -4). The solution claims the normal is (-1, 0, -2), which is a valid scaling of (-2, 0, -4). However, the constant term calculation is wrong. n . P = (-1)(1) + (0)(-2) + (-2)(1) = -1 - 2 = -3. The equation is -x - 2z = -3. Wait, let me re-read the solution's vectors. Line 1 says PQ is (-2, 3, 1). Line 2 says PR is (-2, 5, 1). Line 3 says n is (-1, 0, -2). Let's check the cross product of (-2, 3, 1) and (-2, 5, 1). i(3-5) - j(-2+2) + k(-10+6) = -2i - 0j - 4k = (-2, 0, -4). Scaling by 1/2 gives (-1, 0, -2). This is correct. Line 4 says n . P = -3. (-1)(1) + 0(-2) + (-2)(1) = -3. This is correct. The final equation is -x - 2z = -3. Why did I think it was wrong? Let me check the points in the final equation. (1, -2, 1): -1 - 2(1) = -3. Correct. (-1, 1, 2): -(-1) - 2(2) = 1 - 4 = -3. Correct. (-1, 3, 2): -(-1) - 2(2) = 1 - 4 = -3. Correct. The solution is actually correct. Why did I flag it? I misread the vectors or the cross product in my head. Let me look closer at the provided solution text. Line 1: PQ. Line 2: PR. Line 3: n. Line 4: constant. Line 5: equation. Everything seems algebraically consistent and correct. Wait, looking at the JSON output format, I need to be adversarial. Is there a subtle error? The problem asks for 'an equation'. The solution provides one. The vectors are correct. The cross product is correct. The dot product is correct. The final equation is correct. I will mark it as pass.
gpt-oss:20b: pass 2026-09-27
qwen3.6:27b-mlx: fail (error) 2026-09-26 — The vector PR is calculated incorrectly; P=(-1,1,2) and R=(-1,3,2) yield PR=<0, 2, 0>, but the solution uses <-2, 5, 1>. Consequently, the normal vector and the final equation are wrong.
gpt-oss:20b: pass 2026-09-26

Proved: SymPy reduced the difference between the two sides to zero. Checked independently: a separate method, named above, confirmed it. Checked numerically: the two sides agree at every sampled point, which is evidence, not proof. Reviewed: a model or a person read it; that is all a sentence can have. Solution by generator:structured/plane_through_points, checked 2026-10-13 with SymPy 1.14.0.